Hmmm...
We have HOW many Patriarchs of Antioch at the moment, iOTL?
https://en.wikipedia.org/wiki/Patriarch_of_Antioch said:
In the 18th century, succession disputes in the Greek Orthodox and Syriac Orthodox Churches of Antioch led to factions of those churches entering into communion with Rome under claimants to the patriarchate: the Melkite Greek Catholic Patriarch of Antioch and the Syriac Catholic Patriarch of Antioch, respectively. Their Orthodox counterparts are the Greek Orthodox Patriarch of Antioch and the Syriac Orthodox Patriarch of Antioch, respectively.
So 4, of which 2 are part of the Roman Catholic Church (oops).
Given that state of affairs, if the Byzantines controlled Rome, they might well impose their own Orthodox Bishop/Patriarch of Rome, while the Catholics kept an underground church with their own pope.
If some years later, Italy is conquered by barbarians, but the Orthodox have a stronger presence, you COULD get competing Catholic and Orthodox Bishops of Rome.
The biggest problem with this is that the schism hadn't happened yet at any time Byzantium was in any position to conquer Italy; and, in any case, Catholicism and Orthodoxy are very, very similar doctrinally. Not like the difference between Duophysites (Catholic and Orthodox) and Monophysites e.g. the Syriac Church.
To really do this in a plausible fashion, we need a rather stranger history. A couple of proposals
1) Maybe Arianism takes hold in one of Byzantium or Rome, or Donatism in Rome, or some such early on. Then there would be definite doctrinal differences, and a Byzantine conquest of Rome might have resulted in two popes, and described above.
2) The conquest of Italy happens under the Ottomans (or other Muslim power), who install a tame Orthodox Pope under the authority of their tame Patriarch of Constantinople. Catholics hold to their own underground church, less because the Orthodox are so objectionable, and more because they're puppets of Muslims.
3) Arianism is imposed on Italy by Ostrogoths, who survive, and who impose an Arian Bishop of Rome. The local Catholics keep their own bishop (in union with Byzantium - here the Schism never happens and Catholic and Orthodox remain synonyms).